1.3 Linear Functions
Part A: Basics: Slope and Intercept

Next tutorial: Part B: Finding the Equation of a Line

(This topic is also in Section 1.3 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)

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Q What is a linear function?
A A linear function is one whose graph is a straight line (hence the term "linear").

Q How do we recognize a linear function algebraically?
A As follows:

Linear Function

A linear function is one that can be written in the form

f(x) = mx + b Function formExample: f(x) = 3x - 1     m = 3, b = -1
y = mx + b Equation formExample:y = 3x - 1
where m and b are fixed numbers (the names m and b are traditional).

Example

Here is a partial table of values of the linear function f(x) = 3x - 1. Fill in the missing values and press "Check."

x-4-3-2-101234
y-13-10-7-125

   

Plotting a few of these points gives the following graph.

The Role of b in the equation y = mx + b
Let us look more closely at the above linear function, y = 3x - 1, and its graph, shown above. This linear equation has m = 3 and b = -1.

Notice that that setting x = 0 gives y = -1, the value of b.

Numerically, b is the value of y when x = 0

On the graph, the corresponding point (0, -1) is the point where the graph crosses the y-axis, and we say that b = -1 is the y-intercept of the graph

Graphically, b is the y-intercept of the graph

The Role of m in the equation y = mx + b
Notice from the table that the value of y increases by m = 3 for every increase of 1 in x. This is caused by the term 3x in the formula: for every increase of 1 in x we get an increase of 31 = 3 in y.

Numerically, y increases by m units for every 1-unit increase of x.

On the graph, the value of y increases by exactly 3 for every increase of 1 in x, the graph is a straight line rising by 3 for every 1 we go to the right. We say that we have a rise of 3 units for each run of 1 unit. Similarly, we have a rise of 6 for a run of 2, a rise of 9 for a run of 3, and so on. Thus we see that m = 3 is a measure of the steepness of the line; we call m the slope of the line.

Geometrically, the graph rises by by m units for every 1-unit move to the right; m is the slope of the line.

Here is the graph of y = 0.5x + 2, so that b = 2 (y-intercept) and m = 0.5 (slope).

Notice that the graph cuts the y-axis at b = 2, and goes up 0.5 units for every one unit to the right. Here is a more general picture showing two "generic" lines; one with positive slope, and one wqith negative slope.

Graph of y = mx + b
Positive SlopeNegative Slope

Let y = -1.5x + 4.

Q m = b =  

Q The above answers tell us that, whenever x increases by 1 unit,

Q Similarly, whenever x increases by 2 units,

Q Select the correct graph of y = -1.5x + 4. (Gridlines are spaced one unit apart.)


   


The next example will help you recognize a linear function from numerical values.

Here are two functions specified numerically. One of them is linear, the other is not.

x13579
f(x)-44204576
g(x)-4.5-1.51.54.57.5

Mathematicians traditionally use (delta, the Greek equivalent of the Roman letter D) to stand for "difference," or "change in." For example, we write x to stand for "the change in x."

Let us take another look at the linear equation

Now we know that y increases by 3 for every 1-unit increase in x.
Similarly, y increases by 32 = 6 for every 3-unit increase in x.
. . . .
In general, y increases by 3x units for every x-unit change in x.

Using symbols,

Q How do these changes show up on the graph?
A Here again is the graph of y = 3x - 1 , showing two different choices for x and the associated y.


y = 3x - 1

To summarize:

Slope of a Line

The slope of a line is given by the ratio

    m =y=3x =
    Change in y

    Change in x
    =
    Rise

    Run
    =Slope


Definition of the Slope

For positive m, the graph rises m units for every 1-unit move to the right, and rises y = mx units for every x units moved to the right. For negative m, the graph drops |m| units for every 1-unit move to the right, and drops |m|x units for every x units moved to the right.

Graph of y = mx + b
Positive Slope Negative Slope

Fill in the slopes of the following lines.

m =  
   
m =  
   
m =  
   


Getting Familiar with Slopes
It is useful to be able to estimate the slope of a line "at a glance." In particular, you should know how to recognize lines with slope 0, 1, and -1.


               

Computing the Slope of a Line

Q Two points, say (x1, y1) and (x2, y2), determine a line in the xy-plane. How do we find its slope?
A Look at the following figure.


 

As you can see in the figure, the rise is y = y2 - y1, the change in the y-coordinate from the first point to the second, while the run is x = x2 - x1, the change in the x-coordinate.

To summarize:

Linear Function

Computing the Slope of a Line We can compute the slope m of the line through the points (x1, y1) and (x2, y2) using

    m =
    y

    x
    =
    y2 - y1

    x2 - x1

Examples

1. The slope of the line through (x1, y1) = (1, 3) and (x2, y2) = (5, 11) ism =
y2 - y1

x2 - x1
=
11 - 3

5 - 1
=
8

4
= 2
2. The slope of the line through (-1, 3) and (-1, 5) ism =
y2 - y1

x2 - x1
=
5 - 3

-1 - (-1)
=
2

0
, which is undefined, or infinite.
3. The slope of the line through (-1, 0) and (2, -2) ism =  

Before trying the exercises, you should go on to the next tutorial: Part B: Finding the Equation of a Line.

Last Updated: March, 2006
Copyright © 2001, 2007 Stefan Waner